Due to its clustering and self-exciting properties, the Hawkes process has been used extensively in numerous fields ranging from sismology to finance. Since data is often aquired on regular time intervals, we propose a piece-wise constant model based
on a Discrete-Time Hawkes Process (DTHP). We prove that this discrete-time model converges to the usual continuous-time Hawkes process as the time-step tends to zero.
The Robbins-Monro algorithm is a recursive, simulation-based stochastic procedure to approximate the zeros of a function that can be written as an expectation. It is known that under some technical assumptions, a Gaussian convergence can be establish
ed for the procedure. Here, we are interested in the local limit theorem, that is, quantifying this convergence on the density of the involved objects. The analysis relies on a parametrix technique for Markov chains converging to diffusions, where the drift is unbounded.
We obtain an expansion of the implicit weak discretization error for the target of stochastic approximation algorithms introduced and studied in [Frikha2013]. This allows us to extend and develop the Richardson-Romberg extrapolation method for Monte
Carlo linear estimator (introduced in [Talay & Tubaro 1990] and deeply studied in [Pag{`e}s 2007]) to the framework of stochastic optimization by means of stochastic approximation algorithm. We notably apply the method to the estimation of the quantile of diffusion processes. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost.
We consider a stable driven degenerate stochastic differential equation, whose coefficients satisfy a kind of weak H{o}rmander condition. Under mild smoothness assumptions we prove the uniqueness of the martingale problem for the associated generator
under some dimension constraints. Also, when the driving noise is scalar and tempered, we establish density bounds reflecting the multi-scale behavior of the process.
